We have covered a lot of material quite quickly, with a focus on examples.

Now let’s cover some core features of Python in a more systematic way.

This approach is less exciting but helps clear up some details.

Computer programs typically keep track of a range of data types.

For example, `1.5`

is a floating point number, while `1`

is an integer.

Programs need to distinguish between these two types for various reasons.

One is that they are stored in memory differently.

Another is that arithmetic operations are different

- For example, floating point arithmetic is implemented on most machines by a specialized Floating Point Unit (FPU).

In general, floats are more informative but arithmetic operations on integers are faster and more accurate.

Python provides numerous other built-in Python data types, some of which we’ve already met

- strings, lists, etc.

Let’s learn a bit more about them.

One simple data type is **Boolean values**, which can be either `True`

or `False`

In [ ]:

```
x = True
x
```

We can check the type of any object in memory using the `type()`

function.

In [ ]:

```
type(x)
```

In the next line of code, the interpreter evaluates the expression on the right of = and binds y to this value

In [ ]:

```
y = 100 < 10
y
```

In [ ]:

```
type(y)
```

In arithmetic expressions, `True`

is converted to `1`

and `False`

is converted `0`

.

This is called **Boolean arithmetic** and is often useful in programming.

Here are some examples

In [ ]:

```
x + y
```

In [ ]:

```
x * y
```

In [ ]:

```
True + True
```

In [ ]:

```
bools = [True, True, False, True] # List of Boolean values
sum(bools)
```

Complex numbers are another primitive data type in Python

In [ ]:

```
x = complex(1, 2)
y = complex(2, 1)
print(x * y)
type(x)
```

Python has several basic types for storing collections of (possibly heterogeneous) data.

We’ve already discussed lists.

In [ ]:

```
x = ('a', 'b') # Parentheses instead of the square brackets
x = 'a', 'b' # Or no brackets --- the meaning is identical
x
```

In [ ]:

```
type(x)
```

In Python, an object is called **immutable** if, once created, the object cannot be changed.

Conversely, an object is **mutable** if it can still be altered after creation.

Python lists are mutable

In [ ]:

```
x = [1, 2]
x[0] = 10
x
```

But tuples are not

In [ ]:

```
x = (1, 2)
x[0] = 10
```

We’ll say more about the role of mutable and immutable data a bit later.

Tuples (and lists) can be “unpacked” as follows

In [ ]:

```
integers = (10, 20, 30)
x, y, z = integers
x
```

In [ ]:

```
y
```

You’ve actually seen an example of this already.

Tuple unpacking is convenient and we’ll use it often.

To access multiple elements of a list or tuple, you can use Python’s slice notation.

For example,

In [ ]:

```
a = [2, 4, 6, 8]
a[1:]
```

In [ ]:

```
a[1:3]
```

The general rule is that `a[m:n]`

returns `n - m`

elements, starting at `a[m]`

.

Negative numbers are also permissible

In [ ]:

```
a[-2:] # Last two elements of the list
```

The same slice notation works on tuples and strings

In [ ]:

```
s = 'foobar'
s[-3:] # Select the last three elements
```

Two other container types we should mention before moving on are sets and dictionaries.

Dictionaries are much like lists, except that the items are named instead of numbered

In [ ]:

```
d = {'name': 'Frodo', 'age': 33}
type(d)
```

In [ ]:

```
d['age']
```

The names `'name'`

and `'age'`

are called the *keys*.

The objects that the keys are mapped to (`'Frodo'`

and `33`

) are called the `values`

.

Sets are unordered collections without duplicates, and set methods provide the usual set-theoretic operations

In [ ]:

```
s1 = {'a', 'b'}
type(s1)
```

In [ ]:

```
s2 = {'b', 'c'}
s1.issubset(s2)
```

In [ ]:

```
s1.intersection(s2)
```

The `set()`

function creates sets from sequences

In [ ]:

```
s3 = set(('foo', 'bar', 'foo'))
s3
```

Let’s briefly review reading and writing to text files, starting with writing

In [ ]:

```
f = open('newfile.txt', 'w') # Open 'newfile.txt' for writing
f.write('Testing\n') # Here '\n' means new line
f.write('Testing again')
f.close()
```

Here

- The built-in function
`open()`

creates a file object for writing to. - Both
`write()`

and`close()`

are methods of file objects.

Where is this file that we’ve created?

Recall that Python maintains a concept of the present working directory (pwd) that can be located from with Jupyter or IPython via

In [ ]:

```
%pwd
```

If a path is not specified, then this is where Python writes to.

We can also use Python to read the contents of `newline.txt`

as follows

In [ ]:

```
f = open('newfile.txt', 'r')
out = f.read()
out
```

In [ ]:

```
print(out)
```

```
f = open('insert_full_path_to_file/newfile.txt', 'r')
```

One of the most important tasks in computing is stepping through a sequence of data and performing a given action.

One of Python’s strengths is its simple, flexible interface to this kind of iteration via
the `for`

loop.

Many Python objects are “iterable”, in the sense that they can be looped over.

To give an example, let’s write the file us_cities.txt, which lists US cities and their population, to the present working directory.

In [ ]:

```
%%file us_cities.txt
new york: 8244910
los angeles: 3819702
chicago: 2707120
houston: 2145146
philadelphia: 1536471
phoenix: 1469471
san antonio: 1359758
san diego: 1326179
dallas: 1223229
```

Here %%file is an IPython cell magic.

Suppose that we want to make the information more readable, by capitalizing names and adding commas to mark thousands.

The program below reads the data in and makes the conversion:

In [ ]:

```
data_file = open('us_cities.txt', 'r')
for line in data_file:
city, population = line.split(':') # Tuple unpacking
city = city.title() # Capitalize city names
population = f'{int(population):,}' # Add commas to numbers
print(city.ljust(15) + population)
data_file.close()
```

Here `format()`

is a string method used for inserting variables into strings.

The reformatting of each line is the result of three different string methods, the details of which can be left till later.

The interesting part of this program for us is line 2, which shows that

- The file object
`data_file`

is iterable, in the sense that it can be placed to the right of`in`

within a`for`

loop. - Iteration steps through each line in the file.

This leads to the clean, convenient syntax shown in our program.

Many other kinds of objects are iterable, and we’ll discuss some of them later on.

One thing you might have noticed is that Python tends to favor looping without explicit indexing.

For example,

In [ ]:

```
x_values = [1, 2, 3] # Some iterable x
for x in x_values:
print(x * x)
```

is preferred to

In [ ]:

```
for i in range(len(x_values)):
print(x_values[i] * x_values[i])
```

When you compare these two alternatives, you can see why the first one is preferred.

Python provides some facilities to simplify looping without indices.

One is `zip()`

, which is used for stepping through pairs from two sequences.

For example, try running the following code

In [ ]:

```
countries = ('Japan', 'Korea', 'China')
cities = ('Tokyo', 'Seoul', 'Beijing')
for country, city in zip(countries, cities):
print(f'The capital of {country} is {city}')
```

The `zip()`

function is also useful for creating dictionaries — for
example

In [ ]:

```
names = ['Tom', 'John']
marks = ['E', 'F']
dict(zip(names, marks))
```

If we actually need the index from a list, one option is to use `enumerate()`

.

To understand what `enumerate()`

does, consider the following example

In [ ]:

```
letter_list = ['a', 'b', 'c']
for index, letter in enumerate(letter_list):
print(f"letter_list[{index}] = '{letter}'")
```

We can also simplify the code for generating the list of random draws considerably by using something called a *list comprehension*.

List comprehensions are an elegant Python tool for creating lists.

Consider the following example, where the list comprehension is on the right-hand side of the second line

In [ ]:

```
animals = ['dog', 'cat', 'bird']
plurals = [animal + 's' for animal in animals]
plurals
```

Here’s another example

In [ ]:

```
range(8)
```

In [ ]:

```
doubles = [2 * x for x in range(8)]
doubles
```

Many different kinds of expressions evaluate to one of the Boolean values (i.e., `True`

or `False`

).

A common type is comparisons, such as

In [ ]:

```
x, y = 1, 2
x < y
```

In [ ]:

```
x > y
```

One of the nice features of Python is that we can *chain* inequalities

In [ ]:

```
1 < 2 < 3
```

In [ ]:

```
1 <= 2 <= 3
```

As we saw earlier, when testing for equality we use `==`

In [ ]:

```
x = 1 # Assignment
x == 2 # Comparison
```

For “not equal” use `!=`

In [ ]:

```
1 != 2
```

Note that when testing conditions, we can use **any** valid Python expression

In [ ]:

```
x = 'yes' if 42 else 'no'
x
```

In [ ]:

```
x = 'yes' if [] else 'no'
x
```

What’s going on here?

The rule is:

- Expressions that evaluate to zero, empty sequences or containers (strings, lists, etc.) and
`None`

are all equivalent to`False`

.- for example,
`[]`

and`()`

are equivalent to`False`

in an`if`

clause

- for example,
- All other values are equivalent to
`True`

.- for example,
`42`

is equivalent to`True`

in an`if`

clause

- for example,

We can combine expressions using `and`

, `or`

and `not`

.

These are the standard logical connectives (conjunction, disjunction and denial)

In [ ]:

```
1 < 2 and 'f' in 'foo'
```

In [ ]:

```
1 < 2 and 'g' in 'foo'
```

In [ ]:

```
1 < 2 or 'g' in 'foo'
```

In [ ]:

```
not True
```

In [ ]:

```
not not True
```

Remember

`P and Q`

is`True`

if both are`True`

, else`False`

`P or Q`

is`False`

if both are`False`

, else`True`

Let’s talk a bit more about functions, which are all important for good programming style.

As we discussed in the previous lecture, Python functions are very flexible.

In particular

- Any number of functions can be defined in a given file.
- Functions can be (and often are) defined inside other functions.
- Any object can be passed to a function as an argument, including other functions.
- A function can return any kind of object, including functions.

We already gave an example of how straightforward it is to pass a function to a function.

Note that a function can have arbitrarily many `return`

statements (including zero).

Execution of the function terminates when the first return is hit, allowing code like the following example

In [ ]:

```
def f(x):
if x < 0:
return 'negative'
return 'nonnegative'
```

Functions without a return statement automatically return the special Python object `None`

.

Python has a system for adding comments to functions, modules, etc. called *docstrings*.

The nice thing about docstrings is that they are available at run-time.

Try running this

In [ ]:

```
def f(x):
"""
This function squares its argument
"""
return x**2
```

After running this code, the docstring is available

In [ ]:

```
f?
```

```
Type: function
String Form:<function f at 0x2223320>
File: /home/john/temp/temp.py
Definition: f(x)
Docstring: This function squares its argument
```

In [ ]:

```
f??
```

```
Type: function
String Form:<function f at 0x2223320>
File: /home/john/temp/temp.py
Definition: f(x)
Source:
def f(x):
"""
This function squares its argument
"""
return x**2
```

With one question mark we bring up the docstring, and with two we get the source code as well.

`lambda`

¶
The `lambda`

keyword is used to create simple functions on one line.

For example, the definitions

In [ ]:

```
def f(x):
return x**3
```

and

In [ ]:

```
f = lambda x: x**3
```

are entirely equivalent.

To see why `lambda`

is useful, suppose that we want to calculate $ \int_0^2 x^3 dx $ (and have forgotten our high-school calculus).

The SciPy library has a function called `quad`

that will do this calculation for us.

The syntax of the `quad`

function is `quad(f, a, b)`

where `f`

is a function and `a`

and `b`

are numbers.

To create the function $ f(x) = x^3 $ we can use `lambda`

as follows

In [ ]:

```
from scipy.integrate import quad
quad(lambda x: x**3, 0, 2)
```

Here the function created by `lambda`

is said to be *anonymous* because it was never given a name.

In a previous lecture, you came across the statement

```
plt.plot(x, 'b-', label="white noise")
```

In this call to Matplotlib’s `plot`

function, notice that the last argument is passed in `name=argument`

syntax.

This is called a *keyword argument*, with `label`

being the keyword.

Non-keyword arguments are called *positional arguments*, since their meaning
is determined by order

`plot(x, 'b-', label="white noise")`

is different from`plot('b-', x, label="white noise")`

Keyword arguments are particularly useful when a function has a lot of arguments, in which case it’s hard to remember the right order.

You can adopt keyword arguments in user-defined functions with no difficulty.

The next example illustrates the syntax

In [ ]:

```
def f(x, a=1, b=1):
return a + b * x
```

The keyword argument values we supplied in the definition of `f`

become the default values

In [ ]:

```
f(2)
```

They can be modified as follows

In [ ]:

```
f(2, a=4, b=5)
```

To learn more about the Python programming philosophy type `import this`

at the prompt.

Among other things, Python strongly favors consistency in programming style.

We’ve all heard the saying about consistency and little minds.

In programming, as in mathematics, the opposite is true

- A mathematical paper where the symbols $ \cup $ and $ \cap $ were reversed would be very hard to read, even if the author told you so on the first page.

In Python, the standard style is set out in PEP8.

(Occasionally we’ll deviate from PEP8 in these lectures to better match mathematical notation)

Part 1: Given two numeric lists or tuples `x_vals`

and `y_vals`

of equal length, compute
their inner product using `zip()`

.

Part 2: In one line, count the number of even numbers in 0,…,99.

- Hint:
`x % 2`

returns 0 if`x`

is even, 1 otherwise.

Part 3: Given `pairs = ((2, 5), (4, 2), (9, 8), (12, 10))`

, count the number of pairs `(a, b)`

such that both `a`

and `b`

are even.

Write a function that takes a string as an argument and returns the number of capital letters in the string.

Hint: `'foo'.upper()`

returns `'FOO'`

.

When we cover the numerical libraries, we will see they include many alternatives for interpolation and function approximation.

Nevertheless, let’s write our own function approximation routine as an exercise.

In particular, without using any imports, write a function `linapprox`

that takes as arguments

- A function
`f`

mapping some interval $ [a, b] $ into $ \mathbb R $. - Two scalars
`a`

and`b`

providing the limits of this interval. - An integer
`n`

determining the number of grid points. - A number
`x`

satisfying`a <= x <= b`

.

and returns the piecewise linear interpolation of `f`

at `x`

, based on `n`

evenly spaced grid points `a = point[0] < point[1] < ... < point[n-1] = b`

.

Aim for clarity, not efficiency.

Using list comprehension syntax, we can simplify the loop in the following code.

In [ ]:

```
import numpy as np
n = 100
ϵ_values = []
for i in range(n):
e = np.random.randn()
ϵ_values.append(e)
```

Here’s one possible solution

In [ ]:

```
x_vals = [1, 2, 3]
y_vals = [1, 1, 1]
sum([x * y for x, y in zip(x_vals, y_vals)])
```

This also works

In [ ]:

```
sum(x * y for x, y in zip(x_vals, y_vals))
```

One solution is

In [ ]:

```
sum([x % 2 == 0 for x in range(100)])
```

This also works:

In [ ]:

```
sum(x % 2 == 0 for x in range(100))
```

Some less natural alternatives that nonetheless help to illustrate the flexibility of list comprehensions are

In [ ]:

```
len([x for x in range(100) if x % 2 == 0])
```

and

In [ ]:

```
sum([1 for x in range(100) if x % 2 == 0])
```

Here’s one possibility

In [ ]:

```
pairs = ((2, 5), (4, 2), (9, 8), (12, 10))
sum([x % 2 == 0 and y % 2 == 0 for x, y in pairs])
```

In [ ]:

```
def p(x, coeff):
return sum(a * x**i for i, a in enumerate(coeff))
```

In [ ]:

```
p(1, (2, 4))
```

Here’s one solution:

In [ ]:

```
def f(string):
count = 0
for letter in string:
if letter == letter.upper() and letter.isalpha():
count += 1
return count
f('The Rain in Spain')
```

An alternative, more pythonic solution:

In [ ]:

```
def count_uppercase_chars(s):
return sum([c.isupper() for c in s])
count_uppercase_chars('The Rain in Spain')
```

Here’s a solution:

In [ ]:

```
def f(seq_a, seq_b):
is_subset = True
for a in seq_a:
if a not in seq_b:
is_subset = False
return is_subset
# == test == #
print(f([1, 2], [1, 2, 3]))
print(f([1, 2, 3], [1, 2]))
```

Of course, if we use the `sets`

data type then the solution is easier

In [ ]:

```
def f(seq_a, seq_b):
return set(seq_a).issubset(set(seq_b))
```

In [ ]:

```
def linapprox(f, a, b, n, x):
"""
Evaluates the piecewise linear interpolant of f at x on the interval
[a, b], with n evenly spaced grid points.
Parameters
==========
f : function
The function to approximate
x, a, b : scalars (floats or integers)
Evaluation point and endpoints, with a <= x <= b
n : integer
Number of grid points
Returns
=======
A float. The interpolant evaluated at x
"""
length_of_interval = b - a
num_subintervals = n - 1
step = length_of_interval / num_subintervals
# === find first grid point larger than x === #
point = a
while point <= x:
point += step
# === x must lie between the gridpoints (point - step) and point === #
u, v = point - step, point
return f(u) + (x - u) * (f(v) - f(u)) / (v - u)
```

Here’s one solution.

In [ ]:

```
n = 100
ϵ_values = [np.random.randn() for i in range(n)]
```